Theorem satfrel 33325

Description: The value of the satisfaction predicate as function over wff codes at a natural number is a relation. (Contributed by AV, 12-Oct-2023.)

Assertion

Ref	Expression
satfrel	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))

Proof of Theorem satfrel

Dummy variables 𝑎 𝑖 𝑗 𝑢 𝑣 𝑥 𝑦 𝑧 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6771	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘∅))
2	1	releqd 5689	. . . . 5 ⊢ (𝑎 = ∅ → (Rel ((𝑀 Sat 𝐸)‘𝑎) ↔ Rel ((𝑀 Sat 𝐸)‘∅)))
3	2	imbi2d 341	. . . 4 ⊢ (𝑎 = ∅ → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑎)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘∅))))
4		fveq2 6771	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘𝑏))
5	4	releqd 5689	. . . . 5 ⊢ (𝑎 = 𝑏 → (Rel ((𝑀 Sat 𝐸)‘𝑎) ↔ Rel ((𝑀 Sat 𝐸)‘𝑏)))
6	5	imbi2d 341	. . . 4 ⊢ (𝑎 = 𝑏 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑎)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑏))))
7		fveq2 6771	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘suc 𝑏))
8	7	releqd 5689	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (Rel ((𝑀 Sat 𝐸)‘𝑎) ↔ Rel ((𝑀 Sat 𝐸)‘suc 𝑏)))
9	8	imbi2d 341	. . . 4 ⊢ (𝑎 = suc 𝑏 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑎)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
10		fveq2 6771	. . . . . 6 ⊢ (𝑎 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘𝑁))
11	10	releqd 5689	. . . . 5 ⊢ (𝑎 = 𝑁 → (Rel ((𝑀 Sat 𝐸)‘𝑎) ↔ Rel ((𝑀 Sat 𝐸)‘𝑁)))
12	11	imbi2d 341	. . . 4 ⊢ (𝑎 = 𝑁 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑎)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑁))))
13		relopabv 5730	. . . . 5 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}
14		eqid 2740	. . . . . . 7 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
15	14	satfv0 33316	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})
16	15	releqd 5689	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Rel ((𝑀 Sat 𝐸)‘∅) ↔ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}))
17	13, 16	mpbiri 257	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘∅))
18		pm2.27 42	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑏)) → Rel ((𝑀 Sat 𝐸)‘𝑏)))
19		simpr 485	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → Rel ((𝑀 Sat 𝐸)‘𝑏))
20		relopabv 5730	. . . . . . . . . 10 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}
21		relun 5720	. . . . . . . . . 10 ⊢ (Rel (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}) ↔ (Rel ((𝑀 Sat 𝐸)‘𝑏) ∧ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
22	19, 20, 21	sylanblrc 590	. . . . . . . . 9 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → Rel (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
23	14	satfvsuc 33319	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑏 ∈ ω) → ((𝑀 Sat 𝐸)‘suc 𝑏) = (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
24	23	ad4ant123 1171	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → ((𝑀 Sat 𝐸)‘suc 𝑏) = (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
25	24	releqd 5689	. . . . . . . . 9 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → (Rel ((𝑀 Sat 𝐸)‘suc 𝑏) ↔ Rel (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})))
26	22, 25	mpbird 256	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))
27	26	exp31 420	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑏 ∈ ω → (Rel ((𝑀 Sat 𝐸)‘𝑏) → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
28	27	com23 86	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Rel ((𝑀 Sat 𝐸)‘𝑏) → (𝑏 ∈ ω → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
29	18, 28	syld 47	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑏)) → (𝑏 ∈ ω → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
30	29	com13 88	. . . 4 ⊢ (𝑏 ∈ ω → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑏)) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
31	3, 6, 9, 12, 17, 30	finds 7739	. . 3 ⊢ (𝑁 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑁)))
32	31	com12 32	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑁 ∈ ω → Rel ((𝑀 Sat 𝐸)‘𝑁)))
33	32	3impia 1116	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067  {crab 3070   ∖ cdif 3889   ∪ cun 3890   ∩ cin 3891  ∅c0 4262  {csn 4567  ⟨cop 4573   class class class wbr 5079  {copab 5141   ↾ cres 5592  Rel wrel 5595  suc csuc 6267  ‘cfv 6432  (class class class)co 7271  ωcom 7706  1^st c1st 7822  2^nd c2nd 7823   ↑_m cmap 8598  ∈_𝑔cgoe 33291  ⊼_𝑔cgna 33292  ∀_𝑔cgol 33293   Sat csat 33294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-inf2 9377

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-goel 33298  df-sat 33301

This theorem is referenced by:  satfdmlem  33326  satffunlem1lem2  33361  satffunlem2lem2  33364